Natural porous media, such as subterranean reservoirs containing hydrocarbons, are typically highly heterogeneous and complex geological formations. While recent advances, specifically in characterization and data integration, have provided for increasingly detailed reservoir models, classical simulation techniques tend to lack the capability to honor the fine-scale detail of these structures. Various multi-scale methods have been developed to deal with this resolution gap.
These multi-scale methods, which can be used for simulation of fluid flow in a subterranean reservoir, can be categorized into multi-scale finite-element (MSFE) methods, mixed multi-scale finite-element (MMSFE) methods, and multi-scale finite-volume (MSFV) methods. These methods aim to reduce complexity of the reservoir model by incorporating the fine-scale variation of coefficients into a coarse-scale operator. This is similar to upscaling methods, which target coarse-scale descriptions based on effective, tensorial coefficients; however, multi-scale methods also allow for reconstruction of the fine-scale velocity field from a coarse-scale pressure solution. If a conservative fine-scale velocity field is obtained, which typically the MMSFE and MSFV methods can provide, the velocity field can then be used to solve the saturation transport equations on the fine grid. It will be appreciated by one skilled in that art, that for problems arising from flow and transport in porous media, a conservative velocity is desired for the transport calculations.
These multi-scale methods can be applied to compute approximate solutions at reduced computational cost. The multi-scale solutions can differ from the reference solutions that are computed with the same standard numerical scheme on the fine grid. While the permeability fields characterized by two separable scales typically converge with respect to coarse-grid refinement, these methods may not converge in the absence of scale separation due to error introduced by multi-scale localization assumptions. For instance, error introduced by a multi-scale method, with respect to the coarse cells, is typically prominent in the presence of large coherent structures with high permeability contrasts, such as nearly impermeable shale layers, where no general accurate localization assumption exists.
Multi-scale methods that are based on local numerical solutions of the fine-scale problem and thus honor the provided permeability field can be used to derive transmissibilities for the coarse problem. The quality of multi-scale results depends on the localization conditions employed to solve the local fine-scale problems. Previous methods have employed global information, such as an initial global fine-scale solution, to enhance the boundary conditions of the local problems. However, these methods may not provide value for fluid flow problems with high phase viscosity ratios, frequently changing boundary conditions, or varying well rates. Other methods have iteratively improved the coarse-scale operator. For instance, the adaptive local-global (ALG) upscaling approach is based on global iterations to obtain a self-consistent coarse-grid description. Recently, ALG was also employed to improve the local boundary conditions in the multi-scale finite volume element method (ALG-MSFVE). While the ALG method has shown to be more accurate than local upscaling methods and leads to asymptotic solutions for a large number of iterations, the solutions typically can be different from standard fine-scale solutions and the error due to ALG can be problem dependent.